# 1+2+3+4+...=-1/12? proof

by fargoth
Tags: 1 or 12, proof
 P: 400 I've see this neat proof: http://www.youtube.com/watch?v=E-d9m...ion_3085392237 (for some reason the youtube tag didn't work in preview...) And now I don't see how what I've learned about series convergence is true... I've been told that if $a_n > b_n \forall n$ then $\sum a_n > \sum b_n$ therefore, if $\sum b_n$ is divergent then, $\sum a_n$ must be too. Also, If the partial sum diverges, the series is said to be divergent, isn't it? And what about $a_n \neq 0$ for n that tends to infinity? So many ways I could show this series diverges, yet he show it's equal to -1/12??? Where am I, or is he, wrong?
 P: 18 I think this cannot be true. The sum of all natural numbers up to N equals (as also shown in the end of the video) ## N(N+1)/2 ##. This obviously goes to infinity as N goes to infinity. And of course there is also no way how strictly positive numbers can add up to give a negative result.
 P: 400 That's what I was saying :) So where is he wrong?
P: 502